Optimal blind carrier recovery for MPSK burst transmissions

The paper introduces and analyzes the asymptotic (large sample) performance of a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of carrier phase, frequency offset, and Doppler rate for burst-mode phase-shift keying transmissions. An optimal or "matched" nonlinear estimator that exhibits the smallest asymptotic variance within the family of envisaged blind NLS estimators is developed. The asymptotic variance of these estimators is established in closed-form expression and shown to approach the Cramer-Rao lower bound of an unmodulated carrier at medium and high signal-to-noise ratios (SNR). Monomial nonlinear estimators that do not depend on the SNR are also introduced and shown to perform similarly to the SNR-dependent matched nonlinear estimator. Computer simulations are presented to corroborate the theoretical performance analysis.     

A matched nonlinearity for phase estimation of a PSK- modulated carrier (Corresp.)

A nonlinear phase estimator for a phase-shift keyed (PSK)-modulated carrier has been developed by Viterbi and Viterbi. Their analysis is extended, and an optimal or "matched" nonlinearity is derived for the estimator.     

Sequential sequence estimation for channels with intersymbolinterference of finite or infinite length

A sequential sequence estimator for channels with finite or infinite intersymbol interference (ISI) is proposed. It consists of a whitened matched filter followed by a sequential algorithm. A key component of a sequential estimator is a suitable metric that accounts for the unequal search depth. This metric is derived for the ISI channel. Computer simulations for three different channels show that the sequential sequence estimator has considerably less computational complexity than a maximum-likelihood sequence estimator using the Viterbi algorithm (MLSE-VA), while still maintaining the optimum estimation performance of the MLSE-VA, a performance not achievable with any reduced-state Viterbi algorithm     

Nonparametric density estimation for classes of positive randomvariables

A kernel-based density estimator for positive random variables is proposed and analyzed. In particular, a nonparametric estimator is developed which takes advantage of the fact that positive random variables can be represented as the norms of random vectors. By appropriately choosing the dimension of the assumed vector space, the estimator can be structured to exploit a priori knowledge about the density to be estimated. The asymptotic properties (e.g., pointwise and L₁-consistency) of this density estimator are investigated and found to be similar to the desirable features of the standard kernel estimator. An upper bound on the expected value of the L₁ error is also derived which provides insight into the behavior of the estimator. Upon using this upper bound, the optimal form for the estimator (i.e., the kernel function, the smoothing factor, etc.) is selected via a minimax strategy. In addition, this upper bound is used to compare the asymptotic performance of the proposed estimator to that of the standard kernel estimator and to boundary-corrected kernel estimators. Numerical examples illustrate that the proposed scheme outperforms the standard and boundary-corrected estimators for a variety of density types     

Asymptotically optimum estimation of signal waveform in the presence of uncertainties about the steering vector

The problem of signal waveform estimation using an antenna array in case of uncertainties about the steering vector is considered. New asymptotically (in sample size) optimum estimators are derived. In contrast to the known optimal solutions, the employed method of synthesis yields non-iterative direct-form estimators. Simulation results are provided to evaluate the performance of the synthesized estimators. It is shown that the proposed asymptotic estimators perform as well as the iterative optimal estimators and they outperform the MVDR estimator in a wide range of input signal-to-noise and uncertainty ratios.     

Blind feedforward cyclostationarity-based timing estimation for linear modulations

By exploiting a general cyclostationary (CS) statistics-based framework, this letter develops a rigorous and unified asymptotic (large sample) performance analysis setup for a class of blind feedforward timing epoch estimators for linear modulations transmitted through time nonselective flat-fading channels. Within the proposed CS framework, it is shown that several estimators proposed in the literature can be asymptotically interpreted as maximum likelihood (ML) estimators applied on a (sub)set of the second- (and/or higher) order statistics of the received signal. The asymptotic variance of these ML estimators is established in closed-form expression and compared with the modified Crame/spl acute/r-Rao bound. It is shown that the timing estimator proposed by Oerder and Meyr achieves asymptotically the best performance in the class of estimators which exploit all the second-order statistics of the received signal, and its performance is insensitive to oversampling rates P as long as P/spl ges/3. Further, an asymptotically best consistent estimator, which achieves the lowest asymptotic variance among all the possible estimators that can be derived by exploiting jointly the second- and fourth-order statistics of the received signal, is also proposed.     

Parameter estimation for random amplitude chirp signals

We consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise, i.e., whose amplitude is randomly time-varying. Two methods for solving this problem are presented. First, an unstructured nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional (2-D) maximization problem. A theoretical analysis of the NLS estimator is presented, and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance that is very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HBF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of the HAF-based estimator is also carried out, and closed-form expressions are derived for the asymptotic variance of the HAF estimators based on the data and on the squared data. Numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods     

An alternative blind feedforward symbol timing estimator using two samples per symbol

Recently, S.J. Lee proposed a blind feedforward symbol timing estimator that exhibits low computational complexity and requires only two samples per symbol (see IEEE Commun. Lett., vol.6, p.205-7, 2002). We analyze Lee's estimator rigorously by exploiting efficiently the cyclostationary statistics present in the received oversampled signal; its asymptotic (large sample) bias and mean-square error (MSE) are derived in closed-form expression. A new blind feedforward timing estimator that requires only two samples per symbol and presents the same computational complexity as Lee's estimator is proposed. It is shown that the proposed new estimator is asymptotically unbiased and exhibits smaller MSE than Lee's estimator. Computer simulations are presented to illustrate the performance of the proposed new estimator with respect to Lee's estimator and existing conventional estimators.     

Complexity reduction for a multisensor Viterbi equaliser

A multisensor equaliser based on the Viterbi algorithm is presented. The equaliser consists of a multisensor Viterbi estimator and adaptive channel estimators. Its complexity is described and its performance over mobile channels is analysed. It is concluded that the multisensor Viterbi equaliser is capable of considering truncated channels, thereby allowing a considerable reduction in complexity     

Simple coherent receivers for partial response continuous phasemodulation

By using a pulse-amplitude-modulation representation of binary continuous-phase-modulation signals, the authors develops a novel optimum Viterbi sequence detector and a near-optimum Viterbi receiver with low complexity. For modulation index 0.5, where a linear receiver can be used, a minimum-mean-squared-error linear receiver filter is derived. The performance of all of these is analyzed, using the Gaussian minimum-shift-keying signal (GMSK) for illustration. It is shown that a GMSK receiver consisting of two matched filters and a four-state Viterbi algorithm performs with less than 0.24-dB degradation compared with the optimal receiver. The linear receiver is optimum for all values of E _b/N₀ (bit-energy-to-noise one-sided spectral density ratio). A design method for its filter is given. The filter is equivalent to a cascade of a matched filter and a Wiener filter estimator. Both upper and lower bounds for the bit-error probability are calculated. Simulation results which confirm the analysis are given     

Maximum Likelihood Receiver for Multiple Channel Transmission Systems

A maximum likelihood (ML) estimator for digital sequences disturbed by Gaussian noise, intersymbol interference (ISI) and interchannel interference (ICI) is derived. It is shown that the sampled outputs of the multiple matched filter (MMF) form a set of sufficient statistics for estimating the input vector sequence. Two ML vector sequence estimation algorithms are presented. One makes use of the sampled output data of the multiple whitened matched filter and is called the vector Viterbi algorithm. The other one is a modification of the vector Viterbi algorithm and uses directly the sampled output of the MMF. It appears that, under a certain condition, the error performance is asymptotically as good as if both ISI and ICI were absent.     

Parameter estimation for Middleton Class A interference processes

The problem of estimating the parameters of the Middleton Class A interference model is considered. On the assumption of the availability of a set of independent samples from Class A envelope distribution, the asymptotic performances of several estimation procedures are explored. From this analysis, estimates based on the method of moments are seen to be consistent and computationally desirable but highly inefficient, whereas more efficient likelihood-based estimators are seen to be computationally unwieldy. However, an estimator that initiates likelihood iteration with the method-of-moments estimates is seen to overcome these difficulties in its asymptotic performance. Unfortunately, simulation of this third estimator for practical sample sizes reveals poor performance under these conditions. To overcome this lack of small-sample efficiency, a similar estimator that initiates likelihood iteration with physically motivated (but nonoptimal) estimates is also proposed. Simulation of this latter estimator for practical sample sizes indicates that near-optimal performance is attained by this technique     

Unified analysis of a class of blind feedforward symbol timing estimators employing second-order statistics

In this letter, all the previously proposed digital blind feedforward symbol timing estimators employing second-order statistics are casted into a unified framework. The finite sample mean-square error (MSE) expression for this class of estimators is established. Simulation results are also presented to corroborate the analytical results. It is found that the feedforward conditional maximum likelihood (CML) estimator and the square law nonlinearity (SLN) estimator with a properly designed prefilter perform the best and their performances coincide with the asymptotic conditional Cramer-Rao bound (CCRB), which is the performance lower bound for the class of estimators under consideration.     

Minimum variance in biased estimation: bounds and asymptotically optimal estimators

We develop a uniform Cramer-Rao lower bound (UCRLB) on the total variance of any estimator of an unknown vector of parameters, with bias gradient matrix whose norm is bounded by a constant. We consider both the Frobenius norm and the spectral norm of the bias gradient matrix, leading to two corresponding lower bounds. We then develop optimal estimators that achieve these lower bounds. In the case in which the measurements are related to the unknown parameters through a linear Gaussian model, Tikhonov regularization is shown to achieve the UCRLB when the Frobenius norm is considered, and the shrunken estimator is shown to achieve the UCRLB when the spectral norm is considered. For more general models, the penalized maximum likelihood (PML) estimator with a suitable penalizing function is shown to asymptotically achieve the UCRLB. To establish the asymptotic optimality of the PML estimator, we first develop the asymptotic mean and variance of the PML estimator for any choice of penalizing function satisfying certain regularity constraints and then derive a general condition on the penalizing function under which the resulting PML estimator asymptotically achieves the UCRLB. This then implies that from all linear and nonlinear estimators with bias gradient whose norm is bounded by a constant, the proposed PML estimator asymptotically results in the smallest possible variance.     

Optimal ARMA parameter estimation based on the sample covariancesfor data with missing observations

The problem of spectral estimation through the autoregressive moving-average (ARMA) modeling of stationary processes with missing observations is considered. A class of estimators based on the sample covariances is presented, and an asymptotically optimal estimator in this class is proposed. The proposed algorithm is based on a nonlinear-least-squares fit of the sample covariances computed from the data to the true covariances of the assumed ARMA model. The statistical properties of the algorithm are explored and used to show that it is asymptotically optimal, in the sense of achieving the smallest possible asymptotic variance. The performance of the algorithm is illustrated by some numerical examples     

A Blind Frequency Offset Estimator for Coherent M-PSK System in Autonomous Radio

This paper presents a novel blind frequency offset estimator for coherent M-PSK systems in an autonomous radio. The proposed estimator is based on the spectrum of the signal’s argument. A data removal block is developed. We derive the distribution of the instantaneous phase, which is applied to indicate that the proposed estimator can be considered as a class of nonlinear least-squares estimator. We provide a method to analyze the asymptotic performance of the proposed estimator. This enable us to predict the mean-square error on frequency offset estimation for all signal-to-noise ratio (SNR) values. Computer simulations indicate that the proposed estimator achieves better performance than the original estimator. The performance of the proposed estimator as a blind estimator is also illustrated.     

Threshold-Based Time-of-Arrival Estimators in UWB Dense Multipath Channels

The need for accurate positioning has gained significant interest recently, especially in cluttered environments where signals from satellite navigation systems are not reliable. Positioning systems based on ultrawide bandwidth (UWB) technology have been considered for these environments because UWB signals are able to resolve multipath and penetrate obstacles. These systems usually obtain range measurements from timeof- arrival (TOA) estimation of the first path, which can be a challenge in dense multipath environments. In this paper, we analyze and compare the performance of matched filter (MF) and energy detector (ED) TOA estimators based on thresholding in UWB dense multipath channels. The main advantage of threshold-based estimators is that they have the potential for complete analog implementation and hence they are particularly attractive for applications that require low cost battery-powered devices. Closed-form expressions for the estimator bias and mean square error (MSE) are derived as a function of the signalto- noise ratio. A comparison with results obtained from Monte Carlo simulation confirms the validity of our analytical approach. This analysis enables us to determine the threshold value that minimizes the MSE, a critical parameter for optimal estimator design. A simple criteria to determine the threshold value is also presented. It is shown that the estimation accuracy is mainly affected by the ambiguity in the selection of the correct peak at the output of the MF or ED, caused by the fading characteristics of the first path. We also evaluate the performance loss of ED estimators with respect to MF estimators. Finally, results based on experimental measurements in an indoor residential environment are presented in order to compare the performance of TOA estimators in realistic environments.     

Low complexity channel estimation for space-time coded wideband OFDM systems

In this article, channel estimation for space-time coded orthogonal-frequency division multiplexing (OFDM) systems is considered. By assuming that the channel frequency response is quasi-static over two consecutive OFDM symbols, we develop channel parameter estimators based on the use of space-time block coded (STBC) training blocks. Using an STBC training pattern, a low-rank Wiener filter-based channel estimator with a significant complexity reduction is proposed. A simplified approach for the optimal low-rank estimator is also proposed to further reduce the estimator complexity while retaining an accurate frequency domain channel estimation. Numerical results are provided to demonstrate the performance of the proposed low complexity channel estimators for space-time trellis coded OFDM systems.     

Asymptotic analysis of blind cyclic correlation-based symbol-rate estimators

This paper considers the problem of blind symbol rate estimation of signals linearly modulated by a sequence of unknown symbols. Oversampling the received signal generates cyclostationary statistics that are exploited to devise symbol-rate estimators by maximizing in the cyclic domain a (possibly weighted) sum of modulus squares of cyclic correlation estimates. Although quite natural, the asymptotic (large sample) performance of this estimator has not been studied rigorously. The consistency and asymptotic normality of this symbol-rate estimator is established when the number of samples N converges to infinity. It is shown that this estimator exhibits a fast convergence rate (proportional to N/sup -3/2/), and it admits a simple closed-form expression for its asymptotic variance. This asymptotic expression enables performance analysis of the rate estimator as a function of the number of estimated cyclic correlation coefficients and the weighting matrix. A justification for the high performance of the unweighted estimator in high signal-to-noise scenarios is also provided.